There isn't anything different between the T-Mobile and Verizon Wireless versions of the update. In fact, Motorola used the original Verizon update post and simply added on a bit about how "Kit Kat is now rolling out to Moto X users on T-Mobile." That said, if you happen to be a Moto X user with T-Mobile -- you have the standard two options ahead of you.

You can sit back, wait for the update notification to arrive and then simply follow the prompts. Alternatively, you can grab your phone and head to Settings -> About phone -> System updates to try and get the process started. Regardless of which path you choose, the update will bring you to 140.44.3.ghost_row.Retail.en.US and with the same in terms of changes, improvements and new features.

The Kit Kat portion means an updated status and navigation bar, new full-screen mode, color emoji support, improved closed captioning support, stronger security and smarter power use. There was also mention of "more tools and capabilities for better app development."

Otherwise, the remaining items breakdown as follows;

Phone dialer: Improved the phone app with the ability to look up contacts directly from the dial pad, see and tap frequent contacts, and search your corporate directory easily.

Camera - Focus and exposure: Enhanced the "touch to focus" option with a new circular, on screen control that can be dragged by your finger around the viewfinder to adjust a photo's focus and exposure.

Hangouts - SMS/MMS support: Incorporated a new version of Google Hangouts that supports integrated SMS/MMS messaging. Hangouts can be set as the default SMS app under Settings > Wireless > Default SMS app.